The radiative transfer equation is one of the most fundamental relationships for describing macroscopic thermal radiation transfer processes. Based on this, the main contribution of this paper is to provide new trends into an unbounded block 3 & times;3 operator matrix defined with a non-maximal domain, offering new insights into the phenomena of radiative transport equations. More precisely, we formulate sufficient and necessary conditions for the entries of the above mentioned model: of the operator matrix to provide an explicit formula for its associated Frobenius-Schur decomposition. Some new spectral properties of such matrix model are also derived to relate its eigenvalues to their two associated Schur complements. As an illustrative example of our findings, we state a model of 3 & times;3 groups of radiative transfer equations in a channel with specific boundary conditions. Our findings build upon and enhance the recent advancements of Bouzidi et al. (Georgian Mathematical Journal, 32, 1, (2025), 37-50). In particular, this study expands the theoretical scope to the realm of 3 & times;3 operator matrix form with a non-maximum domain.
In this study, we examine the fundamental problem of determining the penetration depth of light in a turbid medium, a topic of considerable relevance in optical physics and biomedical applications. To address this issue, the radiative transfer equation (RTE) is employed as a rigorous theoretical framework for describing photon transport in media characterized by both scattering and absorption. The RTE is solved by incorporating the Henyey-Greenstein (HG) phase function, which provides an accurate representation of the angular scattering distribution of photons within the medium. For the solution procedure, the spherical harmonics-based P-N method is adopted due to its proven capability in treating complex scattering behaviors with satisfactory accuracy. The average penetration depth of light is evaluated under various optical parameter configurations to elucidate the influence of scattering characteristics on photon propagation. The computed penetration depths are systematically reported in tabular form, accompanied by a representative figure, corresponding to an approximation order of N = 9. The findings presented herein contribute to a deeper understanding of light-matter interactions in turbid media and provide a useful reference for future theoretical investigations and practical applications in optical transport phenomena.
This article describes a parallel-in-time implementation of a thermal radiation diffusion code using the Parareal methodology. This work builds on previous Parareal diffusion research by demonstrating that the Parareal method can also work well for diffusion problems with non-linear coupling, as thermal radiation does to material energy. Various 1-D Gray and multi-group test problems are shown to demonstrate the efficacy of the Parareal technique. Convergence studies show the relative speed-ups over serial solutions that can be obtained for various degrees of parallelism and choices of coarse and fine solvers. Increasing theoretical algorithmic speed-ups of three times or more were observed on all test problems with greater parallelism and relatively cheaper coarse solvers, indicating favorable scaling of the method to larger problems.
Nonclassical neutral particle transport theory provides a generalized framework for modeling transport in complex, heterogeneous, and correlated media. In the deterministic setting, the classical linear Boltzmann equation is extended to the nonclassical transport equation (NTE), which incorporates nonexponential free-path distributions and memory effects, enabling the representation of a broader range of transport phenomena. The Nonclassical Spherical Harmonic Approximation (NSHA), derived from the NTE through spherical harmonic expansions analogous to the classical PN equations, offers a more efficient direction-of-motion formulation than the full phase-space NTE. A central challenge in applying the NSHA is the consistent formulation of boundary conditions under nonclassical assumptions, since classical conditions such as Marshak, Mark, Asymptotic/Variational (A-V), and Federighi-Pomraning (F-P) must be rederived to reflect nonexponential free-path statistics. This work provides a comprehensive derivation of these approximate boundary conditions for the NSHA in slab geometry, establishing a unified basis for nonclassical transport modeling. Numerical results are presented to evaluate the influence of each boundary condition on the accuracy of NSHA solutions across a range of transport scenarios. Among the conditions examined, the Mark boundary condition consistently produced the most accurate NSHA solutions over the scattering ratios and approximation orders considered.
This study investigates the criticality problem using deep learning techniques. Critical thickness values for slab geometry were computed using the Mathematica code for the P-13 iteration of the P-N method and utilized for training an Artificial Neural Network (ANN) implemented in PyTorch. The dataset includes systems with Mie scattering, represented by a Legendre expansion for anisotropic scattering, as well as varying secondary neutron numbers (c) and anisotropic scattering coefficients. The predicted results are in strong agreement with the calculated critical thickness values. Once trained, the ANN provides rapid predictions, significantly reducing computational effort and time. This approach enables efficient generation of benchmark data and enhances the accessibility of criticality results for nuclear reactor analysis.
We prove a theorem relating the variance of path length estimators for single-collision Monte Carlo source iteration to a parameter that becomes infinitesimally small in an important physical regime arising in radiative transfer. In our usage, "single-collision Monte Carlo source iteration" refers to Monte Carlo Boltzmann transport methods in which each Monte Carlo particle history includes no more than a single collision, and the physics of multiple scattering is modeled by lagging the scattering source term and iterating until this term converges. Our theorem can be used to construct variance reduction techniques which improve the order of the estimator variance. This enables calculations that would otherwise require impractically large sample sizes to achieve practical estimator uncertainties. We believe this is the first postulation of a theorem relating estimator variance to a limiting case parameter for single-collision Monte Carlo source iteration, and the first proof of such a theorem. We illustrate the theorem's value with an example in which the authors of a transport method used the theorem to design a variance reduction technique that improved the uncertainty of their solution by a factor of about 500 for a proxy problem from radiative transfer that contains both optically-thick and optically-thin material.
Domain decomposition is a technique used to reduce memory overhead on large neutron transport problems. During the Center for Exascale Monte Carlo Neutron Transport project, this technique was developed in two major thrusts: processor allocation, and reproducibility. Processor allocation on high performance computing systems was improved through the development of a machine learning model, which can predict the computation load of a decomposed problem's subdomains quickly and accurately. Reproducibility was improved via a new domain decomposition methodology, which takes advantage of a particle-tracked random number generation method to ensure domain-decomposed simulations generate identical results to non-decomposed simulations. Applications of both of these methodologies in the Monte Carlo Dynamic Code solver are discussed, with a focus on the 3D C5G7 eigenvalue problem.
In McClarren and Gentile, the authors presented a general solution for the radiation intensity in front of a purely absorbing slab moving through vacuum toward an observer at constant speed and with a constant temperature. This test problem demonstrated that ignoring certain material motion correction terms in the transport equation can lead to 20-80% errors in the spectral radiation energy density seen by the observer. Because the test problem involved a vacuum region (meaning zero absorption and scattering opacity), the test problem could not be run by many radiation transport algorithms. All simulations presented in (McClarren and Gentile) used Implicit Monte Carlo. In this work we will describe how to modify the solution so that the region between the slab and the observer has a nonzero opacity, and will present results for the modified test problem with multigroup diffusion and SN as well as IMC.
This study investigates the impact of the sample's intrinsic temperature on the line-scan profile of a silicon trapezoidal structure by employing Monte Carlo simulations with a focusing electron beam. This study first reveals that for a specific trapezoidal line, the line-profile contrast increases with temperature across varying focus depths due to the extra energy gain by electrons from the sample. In addition, the optimal focus depth, associated with minimal bias, decreases as temperature rises. This study demonstrates strong potential to provide valuable insights into linewidth measurement techniques for nanostructures using scanning electron microscopes under more realistic environmental conditions.
Various anisotropic scattering kernels are presented to solve the transport equation in multiplying media, and calculations are also performed for the isotropic scattering case and the non-multiplying medium to enable a meaningful comparison. Henyey-Greenstein-like scattering kernel is employed to model anisotropic scattering of neutrons. The effect of anisotropic scattering on the diffusion length of a multiplying slab is investigated using the Legendre polynomial expansion method. A comprehensive numerical analysis of diffusion lengths was performed to represent the fuel system in a nuclear reactor.
Monte Carlo simulations of neutronic systems are computationally intensive and demand significant memory resources for high-fidelity modeling. Compressed sensing enables accurate reconstruction of signals from significantly fewer samples than traditional methods. The specific implementation of compressed sensing investigated here involves the use of overlapping cells to collect tallies. Increasing the number of samples improves the reconstruction accuracy, although the marginal gains diminish with more samples. Reconstruction quality is strongly influenced by the sparsity parameter used in basis pursuit denoising. Across the three test cases considered, memory reductions of up to 81.25% (96.25%) are demonstrated for 2D (3D) reconstructions, with select scenarios achieving reconstruction errors within 1 standard deviation of the corresponding high-fidelity reference results.
This paper presents a new Monte Carlo (MC) algorithm for time-dependent particle transport problems with global variance reduction based on automatic weight windows (WWs). The centers of WWs at a time step are defined by the solution of an auxiliary hybrid MC / deterministic problem formed by the low-order second-moment (LOSM) equations. The closures for the hybrid LOSM equations are calculated by the MC method. The LOSM equations are discretized by a scheme of the second-order accuracy in time and space. Filtering techniques are applied to reduce noise effects in the LOSM closures. The WWs defined with the auxiliary solution give rise to sufficiently uniform MC particle distribution in space on each time step. The algorithm is analyzed by means of an analytic transport benchmark. We study performance of the MC algorithm depending on a set parameters of WWs. Figure of merit and relative error results are presented, demonstrating the performance of the hybrid MC method and quantifying its computational efficiency.
Discrete ordinates S_N transport solvers on unstructured meshes pose a challenge to scale due to complex data dependencies, memory access patterns and a high-dimensional domain. In this paper, we review the performance bottlenecks within the shared memory parallelization scheme of an existing transport solver on modern many-core architectures with high core counts. With this analysis, we then survey the performance of this solver across a variety of compute hardware. We then present a new Asynchronous Many-Task (AMT) algorithm for shared memory parallelism, present results showing an increase in computational performance over the existing method, and evaluate why performance is improved.
This paper presents multi-level hybrid transport (MLHT) methods for solving the neutral particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multi-level Monte Carlo (MLMC) approach. The general MLMC algorithm is defined by the recursive estimation of the expected value of a solution functional's correction with respect to a neighboring grid. MLMC theory optimizes the total computational cost for estimating a functional to within a target accuracy. The proposed MLHT algorithms are based on the quasidiffusion (Variable Eddington Factor) and second-moment methods. For these methods, the low-order equations for the angular moments of the high-order transport solution are discretized in space. Monte Carlo techniques compute the closures for the low-order equations; then, the equations are solved, yielding a single realization of the global flux solution. The ensemble average of the realizations yields the level solution. The results for 1-D slab transport problems demonstrates weak convergence of the functionals considered. We observe that the variance of the correction factors decreases faster than the increase in computational costs of generating an MLMC sample. In the problems considered, the variance and costs of the MLMC solution are driven by the coarse grid calculations.
In Monte Carlo radiation transport calculations, Woodcock-delta tracking is a common alternative to the more popular surface tracking technique. In this work we introduce a delta-tracking algorithm that tallies fluxes to a structured rectilinear mesh using the track-length estimator. This development also enables hybrid surface-delta tracking algorithms, because the track-length tally can be used everywhere for scalar flux estimation regardless of which tracking algorithm is employed. We use this tallying technique to develop a novel hybrid-in-energy method. We also implement a hybrid-in-material method, like what is implemented in Serpent2. We demonstrate that these delta tracking algorithms can be used in conjunction with continuously moving surfaces. We compare these methods showing figures of merit on four time-dependent problems (multi-group and continuous energy) solved with CPU- and GPU-based computers. Our implementation of delta tracking with a track length tally modestly improves figures of merit compared to standard delta tracking with a collision estimator and surface tracking with a track length estimator (1.5X 2.5X) for a problem with significant void regions. For both multi-group and continuous energy pressurized water reactor benchmarks, standard delta tracking with a collision estimator performs best. Hybrid-in-energy methods show significant improvements (7X-11X) for a continuous energy reactor benchmark problem.
Thermal radiative transfer (TRT) governs phenomena ranging from supernovas in astrophysics to laser-driven fusion experiments in plasma physics. The interaction of radiation and matter involves prohibitively small time scales, nonlinear coupling, and high-dimensional particle dynamics, making conventional numerical methods prohibitively expensive. Dynamical low-rank approximation (DLRA), combined with asymptotic-preserving discretizations, offers a promising direction, but until now its use for nonlinear TRT has been fundamentally limited: stability regions of existing DLRA integrators are unknown in realistic nonlinear regimes, and coefficient updates remain computationally costly. We present an asymptotic-preserving, locally conservative, rank-adaptive, and parallel integrator for a macro-micro decomposition-based DLRA of the nonlinear TRT equations. Unlike previous approaches, our method is provably energy stable in the nonlinear setting, with step-size restrictions that capture both hyperbolic and parabolic CFL conditions. The integrator is constructed from the parallel BUG integrator, thus eliminating the need for augmented coefficient updates. In the setting of the parallel integrator and micro-macro decompositions, we propose a strategy to enforce reflection-transmission type boundary conditions in the low-rank factors. These advances resolve long-standing stability and efficiency obstacles, enabling DLRA to be applied robustly to nonlinear TRT with stability guarantees. Numerical experiments confirm the accuracy and efficiency of the proposed approach.
This paper investigates the development of Radiative Transfer Equations (R.T.E.) under oscillatory time-dependent boundary conditions, extending previous work in (Abdmouleh et al. 2024) that considered reflective boundaries. The introduction of oscillatory boundary forcing enables the modeling of realistic physical situations, such as pulsed radiation sources and periodically varying thermal environments, which cannot be captured under reflective conditions. The study combines theoretic analysis with numerical simulations to examine the influence of time-varying boundary conditions on energy density distributions, and scattering mechanisms. Comparative simulations between reflective and oscillatory time-dependent boundaries reveal that the latter induce sustained oscillations, phase shifts, and resonance effects, offering insights into transient radiation-medium interactions. The results demonstrate that oscillatory time-dependent boundary conditions provide a more versatile framework for understanding radiative transfer phenomena.
